The derivative of arctan(x) is given as follows:
Let y = arctan(x)
Then, x = tan(y)
Differentiating both sides with respect to x, we get:
1 = sec²(y) * dy/dx
dy/dx = 1 / sec²(y)
We know that sec²(y) = 1 + tan²(y)
Substituting tan(y) = x, we get:
sec²(y) = 1 + x²
Substituting this back in the above expression, we get:
dy/dx = 1 / (1 + x²)
Therefore, the derivative of arctan(x) with respect to x is 1 / (1 + x²).
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